Subgradient Method Research Articles

A new subgradient extragradient method for solving the split modified system of variational inequality problems and fixed point problem

We introduce a new subgradient extragradient algorithm utilizing the concept of the set of solutions of the split modified system of variational inequality problems (SMSVIP). Our main theorem is weak convergence theorem for such an algorithm for approximating the fixed point problem in a real Hilbert space. We also apply these results to approximate the split minimization problem. In the last section, we provide an example to illustrate the potential of our main theorem.

Interference Nulling Using Reconfigurable Intelligent Surface

This paper investigates the interference nulling capability of reconfigurable intelligent surface (RIS) in a multiuser environment where multiple single-antenna transceivers communicate simultaneously in a shared spectrum. From a theoretical perspective, we show that when the channels between the RIS and the transceivers have line-of-sight and the direct paths are blocked, it is possible to adjust the phases of the RIS elements to null out all the interference completely and to achieve the maximum <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> degrees-of-freedom (DoF) in the overall <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> -user interference channel, provided that the number of RIS elements exceeds some finite value that depends on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> . Algorithmically, for any fixed channel realization we formulate the interference nulling problem as a feasibility problem, and propose an alternating projection algorithm to efficiently solve the resulting nonconvex problem with local convergence guarantee. Numerical results show that the proposed alternating projection algorithm can null all the interference if the number of RIS elements is only slightly larger than a threshold of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2K(K-1)$ </tex-math></inline-formula> . For the practical sum-rate maximization objective, this paper proposes to use the zero-forcing solution obtained from alternating projection as an initial point for subsequent Riemannian conjugate gradient optimization and shows that it has a significant performance advantage over random initializations. For the objective of maximizing the minimum rate, this paper proposes a subgradient projection method which is capable of achieving excellent performance at low complexity.

Optimization of Subgradient Method Parameters Based on Rank-Two Correction of Metric Matrices

Optimization of Subgradient Method Parameters Based on Rank-Two Correction of Metric Matrices

Proximal Linearized Iteratively Reweighted Algorithms for Nonconvex and Nonsmooth Optimization Problem

The nonconvex and nonsmooth optimization problem has been attracting increasing attention in recent years in image processing and machine learning research. The algorithm-based reweighted step has been widely used in many applications. In this paper, we propose a new, extended version of the iterative convex majorization–minimization method (ICMM) for solving a nonconvex and nonsmooth minimization problem, which involves famous iterative reweighted methods. To prove the convergence of the proposed algorithm, we adopt the general unified framework based on the Kurdyka–Łojasiewicz inequality. Numerical experiments validate the effectiveness of the proposed algorithm compared to the existing methods.

Субградиентные методы для задач негладкой оптимизации с некоторой релаксацией условия острого минимума

In this paper we propose a generalized condition for a sharp minimum, somewhat similar to the inexact oracle proposed recently by Devolder-Glineur-Nesterov. The proposed approach makes it possible to extend the class of applicability of subgradient methods with the Polyak step-size, to the situation of inexact information about the value of the minimum, as well as the unknown Lipschitz constant of the objective function. Moreover, the use of local analogs of the global characteristics of the objective function makes it possible to apply the results of this type to wider classes of problems. We show the possibility of applying the proposed approach to strongly convex non-smooth problems, also, we make an experimental comparison with the known optimal subgradient method for such a class of problems. Moreover, there were obtained some results connected to the applicability of the proposed technique to some types of problems with convexity relaxations: the recently proposed notion of weak $\beta$-quasi-convexity and ordinary quasi-convexity. Also in the paper, we study a generalization of the described technique to the situation with the assumption that the $\delta$-subgradient of the objective function is available instead of the usual subgradient. For one of the considered methods, conditions are found under which, in practice, it is possible to escape the projection of the considered iterative sequence onto the feasible set of the problem.

ON THE STRONG CONVERGENCE FOR ITERATIVE METHODS TO FIND A SOLUTION OF A SPLIT PSEUDOMONOTONE VARIATIONAL INEQUALITY AND A SPLIT FEASIBILITY PROBLEM

In this paper, we introduce two different interative methods for finding a solution of a split pseudomonotone variational inequality and a split feasibility problem in Hilbert spaces. The proposed algorithm is generated based on the subgradient extragradient method which requires only two projections at each iteration step and the second projection is conducted onto the half-space containing the constrained set. The strong convergence is proven with some mild conditions imposed on the operators as well as the parameters. A numerical result is provided at the end of the paper with the use of Python for the convergence illustration purpose of the studied method.

Analysis of Subgradient Extragradient Method for Variational Inequality Problems and Null Point Problems

In this paper, we introduce a new numerical method for finding a common solution to variational inequality problems involving monotone mappings and null point problems involving a finite family of inverse-strongly monotone mappings. The method is inspired by the subgradient extragradient method and the regularization method. Strong convergence results of the proposed algorithms have been obtained under some suitable conditions.

Robust dynamic backstepping averaged sub-gradient integral sliding mode control for navigation of mobile robots

The main goal of the current investigation is to present an output feedback robust control that regulates the motion of mobile robotic autonomous devices based on the application of the dual averaged sub-gradient descendant method. Sub-gradient realization aims to solve the design of a proposed extremum seeking control to stabilize the trajectory tracking error between the three-dimensional position of mobile robotic autonomous devices and some attainable reference trajectories in the same spatial domain. The controller belongs to a class of sliding mode controller with variable gain based on a sliding surface depending on the tracking error and its time derivative. This derivative is estimated with the application of a distributed time-dependent super-twisting robust differentiator. The proposed mobile robotic autonomous device dynamics considers the dynamics of actuators (direct current motors) which drives the mobilization of mobile robotic autonomous devices. The dynamic nature of the mobile robotic autonomous devices regulated by motor actuators induces a backstepping-like controller design based on a two-stage sequential application of the sub-gradient strategy, one for the mobile robotic autonomous devices and the second for the motor dynamics. The mixed strategy appears as a novel formulation to solve the path track for reference trajectories of mobile robotic systems. The proposed control considers the optimization of the tracking error dependent functional without the full knowledge of the mobile robotic autonomous device dynamics. With the aim of proving the effectiveness of the suggested design, two numerical examples are developed. In these examples, the tracking performance, minimization of the proposed functional as well as control evolution are evaluated. The controlled tracking outcomes of the proposed integral sliding mode control are contrasted with the outcomes produced by a set of proportional–derivative and selected sliding mode (first-order type) controllers. The suggested controller shows a superior tracking of the reference trajectory than the other two comparative controllers, using quality measures such as least mean square error and its integral.

A self-adaptive parallel subgradient extragradient method for finite family of pseudomonotone equilibrium and fixed point problems

A self-adaptive parallel subgradient extragradient method for finite family of pseudomonotone equilibrium and fixed point problems

Quasiconvex functions: how to separate, if you must!

"Since quasiconvex functions have convex lower level sets it is possible to minimize them by means of separating hyperplanes. An example of such a procedure, well-known for convex functions, is the subgradient method. However, to nd the normal vector of a separating hyperplane is in general not easy for the quasiconvex case. This paper attempts to gain some insight into the computational aspects of determining such a normal vector and the geometry of lower level sets of quasiconvex functions. In order to do so, the directional di erentiability of quasiconvex functions is thoroughly studied. As a consequence of that study, it is shown that an important subset of quasiconvex functions belongs to the class of quasidifferentiable functions. The main emphasis is, however, on computing actual separators. Some important examples are worked out for illustration."

Space planning considering congestion in container terminal yards

This paper proposes a nonlinear mathematical model for solving the problem of allocating storage space to future container arrivals in container terminal yards by considering the waiting times of trucks in the yard, called stochastic space allocation model (SSAM). Three heuristic procedures based on the Frank-Wolfe algorithm and the subgradient optimization method are proposed to efficiently solve the problem. From the numerical experiment, it is found that the proposed algorithms for SSAM provide solutions with the gaps between the objective values and the optimal values are no greater than 0.69%. It was also found that the best algorithm among three proposed algorithms (FWMP-S) obtained the average 7.89% improvement in the objective value compared with a previous model with a handling capacity constraint instead of considering waiting time (DSAM). It is found that the SSAM reduces the average system time of transporters for ship operation and that of road trucks by 17.39 and 7.14%, respectively, compared with the DSAM. This study proposes the most appropriate heuristic algorithm depending on the problem characteristics of the SSAM.

Perturbed Fenchel duality and first-order methods

We show that the iterates generated by a generic first-order meta-algorithm satisfy a canonical perturbed Fenchel duality inequality. The latter in turn readily yields a unified derivation of the best known convergence rates for various popular first-order algorithms including the conditional gradient method as well as the main kinds of Bregman proximal methods: subgradient, gradient, fast gradient, and universal gradient methods.

Essentials of numerical nonsmooth optimization

Approximately sixty years ago two seminal findings, the cutting plane and the subgradient methods, radically changed the landscape of mathematical programming. They provided, for the first time, the practical chance to optimize real functions of several variables characterized by kinks, namely by discontinuities in their derivatives. Convex functions, for which a superb body of theoretical research was growing in parallel, naturally became the main application field of choice. The aim of the paper is to give a concise survey of the key ideas underlying successive development of the area, which took the name of numerical nonsmooth optimization. The focus will be, in particular, on the research mainstreams generated under the impulse of the two initial discoveries.

On Distributed Nonconvex Optimization: Projected Subgradient Method for Weakly Convex Problems in Networks

The stochastic subgradient method is a widely-used algorithm for solving large-scale optimization problems arising in machine learning. Often these problems are neither smooth nor convex. Recently, Davis et al. [1-2] characterized the convergence of the stochastic subgradient method for the weakly convex case, which encompasses many important applications (e.g., robust phase retrieval, blind deconvolution, biconvex compressive sensing, and dictionary learning). In practice, distributed implementations of the projected stochastic subgradient method (stoDPSM) are used to speed-up risk minimization. In this paper, we propose a distributed implementation of the stochastic subgradient method with a theoretical guarantee. Specifically, we show the global convergence of stoDPSM using the Moreau envelope stationarity measure. Furthermore, under a so-called sharpness condition, we show that deterministic DPSM (with a proper initialization) converges linearly to the sharp minima, using geometrically diminishing step-size. We provide numerical experiments to support our theoretical analysis.

Revisiting subgradient extragradient methods for solving variational inequalities

Revisiting subgradient extragradient methods for solving variational inequalities

An improved inertial extragradient subgradient method for solving split variational inequality problems

An improved inertial extragradient subgradient method for solving split variational inequality problems

A strong convergence theorem for solving pseudo-monotone variational inequalities and fixed point problems using subgradient extragradient method in Banach spaces

&lt;abstract&gt;&lt;p&gt;In this paper, we introduce an algorithm for solving variational inequalities problem with Lipschitz continuous and pseudomonotone mapping in Banach space. We modify the subgradient extragradient method with a new and simple iterative step size, and the strong convergence to a common solution of the variational inequalities and fixed point problems is established without the knowledge of the Lipschitz constant. Finally, a numerical experiment is given in support of our results.&lt;/p&gt;&lt;/abstract&gt;

Market’s static model for software development based on a transport problem with non-fixed time additions

The authors describe the formulation of the market’s discrete-continuous static model for software development based on a transport problem with non-fixed time additions. In contrast to the existing transport problem with fixed cost surcharges, it is proposed to formulate a minimax setting in time with times that may contain a part that proportionate to the volumes of appointments. Thus, the authors come up with the idea of a hybrid formulation of transport problem with fixed cost surcharges and classical transport problem in time. Such problems arise when the total volume of vehicles which have to be used repeatedly on each route is limited, plus the fixed addition that arises taking into account the delay in making logistics decisions. It is shown that the set a discrete-continuous problem can be approximated from above by the classical transport problem in time, which can also be obtained according to the scheme used in the work by Balinski. The authors also describe an exact algorithm of the branch-and-bound method, based on the geometric interpretation of the problem, which decomposes into subproblems on non-empty faces of the polyhedral set of feasible solutions, which areconvex programming problems that can be numerically solved by the subgradient method described in work by Polyak. The calculation of the lower estimates of the criterion proposed in the work is reduced to the same problems. It is shown that the function of the best values of the criterion on the edges is not sub- or supermodular, as a function of a subset of pairs of indices corresponding to the positive values of the transportation volumes, which makes it impossible to use supermodular programming methods. In connection with the latter, the authors also describe the optimal polynomial version of the branch-and-bound method, obtained by analogy with the solution of the multidimensional assignment problem, and a numerical example of its use. The authors describe interpretation of transport problem with non-fixed additions as a generalized assignment problem with non-fixed price discounts, taking into account the difference between the wholesale and retail prices. The authors come up with the ideaof the application for building digital platforms in the software development market for loading tasks for performers.

Market static model for software development based on a transport problem with quadratic cost additions

The authors propose to formulate a market continuous static model for software development based on the transport problem with quadratic cost additions. In contrast to the existing transport problem with fixed cost surcharges, the authors propose a statement that minimizes the sum of transportation costs, which may contain non-fixed additions proportional to the squares of the volumes of appointments. Thus, a quadratic formulation of the transport problem with non-fixed additions is proposed. It is shown that transport problem with quadratic additions can be reduced to a convex programming problem, which can be solved numerically by the subgradient method, or by the conjugate gradient method for the dual problem. The authors describe the interpretation of transport problem with quadratic additions as an assignment problem with non-fixed price discounts, taking into account the difference between the wholesale and retail prices. This makes possible to apply the set task to build digital platforms in the software development market for loading assignments to performers.

Continuous Network Design Using Objective Function Linearization and Subgradient Methods

Continuous Network Design Using Objective Function Linearization and Subgradient Methods